Abstract
The totality of extensional 1-ary connectives distinguishable in a logical framework allowing sequents with multiple or empty (alongside singleton) succedents form a lattice under a natural partial ordering relating one connective to another if all the inferential properties of the former are possessed by the latter. Here we give a complete description of that lattice; its Hasse diagram appears as Figure 1 in §2. Simple syntactic descriptions of the lattice elements are provided in §3; §§4 and 5 give some additional remarks on matrix methods and on alternative terminology. Background: The size of this lattice was underestimated in [3]; some missing cases were noted in [4] in the course of correcting an example from [3] purporting to show the non-distributivity of the lattice. All the ‘missing cases’ (as well as those originally noted) are covered here. (The present discussion is self-contained.)
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Humberstone, I.L. Singulary Extensional Connectives: A Closer Look. Journal of Philosophical Logic 26, 341–356 (1997). https://doi.org/10.1023/A:1004240612163
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DOI: https://doi.org/10.1023/A:1004240612163